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Start with $\tau_0 = \mathcal{T} \cap \mathcal{S}$. For each successor ordinal $\alpha+1$, let $\tau_{\alpha+1}$ be the set of elements in $\tau_\alpha$ whose preimage under multiplication is in $(\t …

Edit: $(\mathbb{Q},+)$ with the order topology is compactly generated (by the set $\{1/n \mid n \in \mathbb{N}\} \cup \{0\}$) but not locally compact. I would say this is the 'easiest' example that a …

Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence (edit: or net) of positi …

Consider an unrestricted wreath product $G = \prod_X A \rtimes B$, where $A$ is a group and $B$ is some subgroup of $\mathrm{Sym}(X)$. I am wondering what the circumstances are under which $\prod_X A …

"Is there an example of a group of td-type which is not locally profinite?" No. This was proved by D. van Dantzig in the 1930s: Van Dantzig, D.: Zur topologischen Algebra. III. Brouwersche und Cant …

I am interested in examples of the following property, where $G$ is a non-discrete locally compact topological group: (*) The open normal subgroups of $G$ have trivial intersection, but $G$ has an op …

One situation where you can apply tdlc group theory in a nontrivial way to discrete groups is if the group $G$ that you are interested in has a commensurated subgroup $H$ (meaning, all conjugates of $ …

Every compactly generated locally compact group is either of polynomial growth, or it has a quotient that is just non-(polynomial growth). The same also works for a number of similar properties in pl …

Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point …

Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense. Here is an attempt to define topo …

OK, here is an attempted answer under the assumption that $G$ is locally compact, which can perhaps be refined to give a general answer for Polish groups. A good reference would still be appreciated …

Given a topological group $G$, a $G$-space is a topological space $X$ equipped with an action of $G$, such that the map $(g,x) \mapsto g.x$ is continuous. The action is distal if no non-diagonal orbi …

I couldn't quite answer the question as posed, but thanks to John Griesmer's hint I managed to get a positive answer with slightly different (and perhaps more 'natural') hypotheses, so I will put some …

I am looking for an example of the following situation: $G$ is an infinite profinite group, with a dense normal subgroup $N$. However $N$ does not contain any non-trivial closed normal subgroup of $ …

As Taras Banakh says, it really depends on the underlying group. Some comments in the direction of having a unique CH group topology (which of course is not the case in general): Profinite groups ar …